Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 288

y

x

FIGURE 7.3 Deviations of the data from the estimated regression model.

As shown in Fig. 7.4 a, this is an inadequate criterion. The “fit” is not unique as all the lines,
not only the short-dashed or long-dashed lines but the solid line as well, all satisfy the condi-
tion of minimizing the sum of the errors. Obviously, the “best” fit is the solid line. However,
any straight line passing through the midpoint (except a perfect vertical line) results in
a minimum value for the sum of the residual because the errors cancel.
The second choice might seem to be logical, minimizing the sum of the absolute values of
the residuals:

minimizing X minimizing X

n n

y i a 0 a 1 x i

j ε i j ¼

i ¼ 1 i ¼ 1

a) b)

y y

Minimizing

p = 2

Minimizing

Increasing

Minimizing

p

x x

FIGURE 7.4 Examples of “best” fit criteria for regression analysis. (a) Minimizing the sum of errors or the sum
of the absolute errors. (b) Minimizing the l p -norm of the errors.